15 ideas
| 15946 | Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine] |
| Full Idea: Cantor's development of set theory began with his discovery of the progression 0, 1, ....∞, ∞+1, ∞+2, ..∞x2, ∞x3, ...∞^2, ..∞^3, ...∞^∞, ...∞^∞^∞..... | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite VIII.2 |
| 3340 | Von Neumann defines each number as the set of all smaller numbers [Neumann, by Blackburn] |
| Full Idea: Von Neumann defines each number as the set of all smaller numbers. | |
| From: report of John von Neumann (works [1935]) by Simon Blackburn - Oxford Dictionary of Philosophy p.280 |
| 3355 | Von Neumann wanted mathematical functions to replace sets [Neumann, by Benardete,JA] |
| Full Idea: Von Neumann suggested that functions be pressed into service to replace sets. | |
| From: report of John von Neumann (works [1935]) by José A. Benardete - Metaphysics: the logical approach Ch.23 |
| 15911 | Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine] |
| Full Idea: Ordinal numbers are generated by two principles: each ordinal has an immediate successor, and each unending sequence has an ordinal number as its limit (that is, an ordinal that is next after such a sequence). | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 22716 | Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone] |
| Full Idea: At age twenty, Von Neumann devised the formal definition of ordinal numbers that is used today: an ordinal number is the set of all smaller ordinal numbers. | |
| From: report of John von Neumann (works [1935]) by William Poundstone - Prisoner's Dilemma 02 'Sturm' | |
| A reaction: I take this to be an example of an impredicative definition (not predicating something new), because it uses 'ordinal number' in the definition of ordinal number. I'm guessing the null set gets us started. |
| 23634 | Accepting the existence of anything presupposes the notion of existence [Reid] |
| Full Idea: The belief of the existence of anything seems to suppose a notion of existence - a notion too abstract, perhaps, to enter into the mind of an infant. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 05) | |
| A reaction: But even a small infant has to cope with the experience of waking up from a dream. I don't see how existence can be anything other than a primitive concept in any system of ontology. |
| 23635 | Truths are self-evident to sensible persons who understand them clearly without prejudice [Reid] |
| Full Idea: Self-evident propositions are those which appear evident to every man of sound understanding who apprehends the meaning of them distinctly, and attends to them without prejudice. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 10) | |
| A reaction: I suspect that there are some truths which are self-evident to dogs. There are also truths which are self-evident to experts, but not to ordinary persons of good understanding. Self-evidence is somewhat contextual. Self-evidence can be empirical. |
| 7631 | Sensation is not committed to any external object, but perception is [Reid] |
| Full Idea: Sensation, by itself, implies neither the conception nor belief of any external object. ...Perception implies a conviction and belief of something external. ...Things so different in their nature ought to be distinguished. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], II.16), quoted by Barry Maund - Perception | |
| A reaction: Maund sees this as the origin of the two-stage view of perception, followed by Chisholm, Evans, Dretske and Lowe. It implies that 'looks', 'tastes', 'sounds' etc. are ambiguous words, having either phenomenal or realist meanings. I like it. |
| 23637 | Primary qualities are the object of mathematics [Reid] |
| Full Idea: The primary qualities are the object of the mathematical sciences. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 17) | |
| A reaction: He spells out this crucial point, which is not so obvious in Locke. The sciences totally rely on the primary qualities, so it is ridiculous to reject the distinction (which Reid accepts). |
| 23638 | Secondary qualities conjure up, and are confused with, the sensations which produce them [Reid] |
| Full Idea: The thought of a secondary quality always carries us back to the sensation which it produces.We give the same name to both, and are apt to confound them. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 17) | |
| A reaction: 'Redness', for example. Reid puts the point very nicely. Secondary qualities are not entirely mental; they pick out features of the world, but are much harder to understand than the primary qualities. The qualia question lurks. |
| 23639 | It is unclear whether a toothache is in the mind or in the tooth, but the word has a single meaning [Reid] |
| Full Idea: If it be made a question whether the toothache be in the mind that feels it, or in tooth that is affected, much might be said on both sides, while it is not observed that the word has two meanings. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 18) | |
| A reaction: I'm glad Reid was struck by the weird phenomenon of the brain apparently 'projecting' a pain into a tooth. Presumably before the brain's role was known, people were unaware of this puzzle. There certainly are not two distinct experiences. |
| 6492 | Reid is seen as the main direct realist of the eighteenth century [Reid, by Robinson,H] |
| Full Idea: Reid is often represented by modern opponents of the empiricists as the outstanding protagonist of direct or naïve realism and common sense in the eighteenth century. | |
| From: report of Thomas Reid (Essays on Intellectual Powers 2: Senses [1785]) by Howard Robinson - Perception 1.6 | |
| A reaction: Robinson does not deny that this is Reid's view. Keith Lehrer is a great fan of Reid. Personally I think direct realism is quite clearly false, so I find myself losing interest in Reid's so-called 'common sense'. |
| 23641 | People dislike believing without evidence, and try to avoid it [Reid] |
| Full Idea: To believe without evidence is a weakness which every man is concerned to avoid, and which every man wishes to avoid. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 20) | |
| A reaction: It seems to be very common, though, for people to believe things on incredibly flimsy evidence, if they find the belief appealing. This is close to Clifford's Principle, but not quite as dogmatic. |
| 23642 | If non-rational evidence reaches us, it is reason which then makes use of it [Reid] |
| Full Idea: If Nature gives us information of things that concern us, by other means that by reasoning, reason itself will direct us to receive that information with thankfulness, and to make the best use of it. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 20) | |
| A reaction: This is more of a claim than an argument, but it is hard to see how anything could even be seen as evidence if some sort of rational judgement has not been made. The clever detective sees which facts are evidence. |
| 23640 | Only mature minds can distinguish the qualities of a body [Reid] |
| Full Idea: I think it requires some ripeness of understanding to distinguish the qualities of a body from the body; perhaps this distinction is not made by brutes, or by infants. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 19) | |
| A reaction: I'm glad the brutes get a mention in his assessment of these questions. I take such thinking to arise from what can be labelled the faculty of abstraction, which presumably only appears in a mature brain. It is second-level thinking. |